No, he doesn't have any adult items because he doesn't really have them yet, but it still shows up so you can make sure you have it.Besides, link had obtained the ocarina as a child, so that's why he has it.

Given the paradoxical nature of time travel, you have to allow some leeway in terms of believability. Plus, it's a game first, and I personally prefer being able to keep items I find, regardless of when I technically found them.